Source: M.W. Evans' handwritten note in

## Key Derivation 6: Lorentz Invariance of the B Cyclic Theorem

This is again simple to prove, and it might be . . . it through the computer. The B cyclic theorem is

B(1)×B(2) = i B(o) B(3)*                                 (1)
et cyclicum

where

B(1) = B(o)/sqr(2) (iij) eiΦ                                 (2)

B(2) = B(o)/sqr(2) (i+ij) eiΦ                                 (3)

B(3) = B(3)* = B(o) k                                 (4)

Simple algebra shows that eq. (1) is ???

e(1)×e(2) = i e(3)*                                 (5)
et cyclicum

where

e(1) = 1/sqr(2) (iij)                                 (6)

B(2) = 1/sqr(2) (i+ij)                                 (7)

e(3) = e(3)* = k                                 (8)

Eq. (5) is equivalent to

e(1)×e(2) = i e(3)*                                 (9)
et cyclicum

which is the cyclic symmetry of the Cartesian frame itself. A vector field is defined by

V = Vx i + Vy j + Vz k                                 (10)

The Lorentz transform is considered as a Lorentz boost in z

V = Vz kV' = Vz' k'                                 (11)

The effect on the unit vector is:

kk' .                                 (12)

However, i and j are unchanged, so

i×j = (i×j)'                                 (13)

and

k = k'                                 (14)

This means that eqns. (1), (5) and (9) are Lorentz invariant. Eq. (1) is

B(1)×B(2) = i B(o) B(3)*                                 (15)

Under the Lorentz boost this is invariant.

Bruhn attempted to deliberately confuse people by asserting otherwise. Hehl tried to compound this confusion.

## Commentary by G.W. Bruhn

The above "Key Derivation 6" is apparently written as a revision of Evans' former "rebuttal", part 1 of

Some nonsense contained in that former "rebuttal" has disappeared now, e.g. his remarks on the value of v (v=0). However, Evans shows his general misunderstanding of the technical details of the Lorentz transform as are available in suitable textbooks, e.g. in J.D. Jackson's Classical Elektrodynamics.

Evidently Evans has no detailed knowledge of the Lorentz transform which shall be displayed here for the reader:

### Lorentz transform of a z-boost K → K'

using the abbreviations β = v/c, γ = (1−β²)−½.

#### 1. Transform of the coordinate differentials dxo = c dt, dx1 = dx, dx2 = dy, dx3 = dz

(L1-0)                                 dxo' = γ (dxo − β dx3)
(L1-1)                                 dx1' = dx1
(L1-2)                                 dx2' = dx2
(L1-3)                                 dx3' = γ (dx3 − β dxo)

2. Contragredient transform of the coordinate base vectors ik = ∂k: io = h, i1 = i, i2 = j, i3 = k.

(L2-0)                                 io' = γ (io + β i3)
(L2-1)                                 i1' = i1
(L2-2)                                 i2' = i2
(L2-3)                                 i3' = γ (i3 + β io)

The reader himself should check the invariance relation ik'dxk' = ikdxk (to be summed over k=0,1,2,3).

On this background we can consider Evans' note from above. We state the following objections:

(i) The objects of a Lorentz transform are 4-(dimensional)vectors, not 3-vectors as given by Evans' eq. (10). A vector with zero 0-component relative to the frame K (the vector considered in (10)) has non-zero 0-component relative to the frame K'.

(ii) Evans asserts the rule (14), k = k', which is wrong because of contradicting eq. (L2-3).

(iii) Evans uses the ×-product which is defined as bilinear antisymmetric mapping × : Ä. It is well-known that × cannot be extended to the 4-dimensional case, to a bilinear antisymmetric mapping × : R1+3ÄR1+3R1+3. Therefore,

### Evans considerations using the operation × are without meaning in the case of SRT.

Especially, Evans failed proving the Lorentz-invariance of his B cyclic theorem once more.

Recommendation to MWE

Evans should critically reread his former publications on the Lorentz invariance of his B cyclic theorem taking the criticisms into account.